{"paper":{"title":"Class of smooth functions in Dirichlet spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Liping Li, Patrick J. Fitzsimmons","submitted_at":"2016-11-21T13:33:31Z","abstract_excerpt":"Given a regular Dirichlet form $(\\mathcal{E},\\mathcal{F})$ on a fixed domain $E$ of $\\mathbb{R}^d$, we first indicate that the basic assumption $C_c^\\infty(E)\\subset \\mathcal{F}$ is equivalent to the fact that each coordinate function $f^i(x)=x_i$ locally belongs to $\\mathcal{F}$ for $1\\leq i\\leq d$. Our research starts from these two different viewpoints. On one hand, we shall explore when $C_c^\\infty(E)$ is a special standard core of $\\mathcal{F}$ and give some useful characterizations. On the other hand, we shall describe the Fukushima's decompositions of $(\\mathcal{E},\\mathcal{F})$ with re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06778","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}